Graphical representations of summary statistics
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An ecologist surveys the age of about 100 trees in a local forest. He uses a box-and-whisker plot to map his data shown below. What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box-and-whisker plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information like, what is the median? And where do most of the ages of the trees sit? So this whisker part, so you could see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest to data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in the years. And it says at the highest-- the oldest tree right over here is 50 years. So if we want the range-- and when we think of range in a statistics point of view we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at-- well, let me explain it to you this way. This line right over here, this is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box-and-whiskers plot tells us that half of the ages of the trees are less than 21 and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all the trees that are less than the real median or less than the main median. So this is in the middle of all of the ages of trees that are less than 21. This is the middle age for all the trees that are greater than 21 or older than 21. And so we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for our first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly a fourth of the tree, because the way you calculate it, sometimes a tree ends up in one point or another, about a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years. And you can even see it. It's closer to the left of the box and closer to the end of the left whisker than the end of the right whisker.
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